Effect flashback in mathematics education
For some reason (documented in the literature of cognitive psychology by Daniel Kahneman), many educators believe that the solution of a mathematical problem (school mathematics) can be discovered by any trainee. But it is easy to see that even the teacher can be difficult to discover (or rediscover). These days
was designing a training module on the use of complex numbers to solve geometric problems. At one point she needed to justify (prove) that multiplication by the complex z = r (cost + Isent) executes two actions on another complex z '(z times): the longer times and tour r t degrees.
And needed proof of this result, in turn, the formulas of sine and cosine of the angle sum. A widely used formulas without anyone wondering why they are valid. Are some formulas in some way and "natural" is nauralizado use in solving problems. But considering that the audience targeted by the training module that concerns me is that of teens interested in math contest, then I myself felt the need to demonstrate these formulas. (Here comes the personal opinion, it is difficult to decide in a situation Teaching what to say and what to keep ... but ...)
And I said, "cakewalk." This should be easy ... But no. The key idea of \u200b\u200bthe show did not come, although he was convinced that it should be easy. So I had to resort to a book. Found appropriate by the Dolciani (Modern Introductory Analysis), a very good school math book despite being seventies (I mean the time of axiomatic fashion.)
I had seen and proved the theorem once, so to see the Dolciani experienced a rediscovery and I felt a little embarrassed with myself. Because the key idea is extremely simple! (It is the classic, "expresses the amount of two different modes, retainers and clear") And yes, it is difficult not to conclude - in these cases - that "anyone can come up with."
The reader may consider the following figure as a test "almost speechless" in the formula of the cosine of the angle sum. Just "see" that PQ = P'Q ', applying the formula of distance between two points and clear.
But I wish to emphasize here is that this feeling of "anyone can come up with is a kind of reverse conclusion:" Now I saw the solution seems trivial, so it was always trivial ".
Let me conclude these reflections with a moral teaching. What seems to be the case, trying to escape the "optical illusions" retrospective effect is that an idea, however simple it may seem in retrospect, may be inaccessible to the learner's cognition without the help of a suggestion from the instructor. But you can also say that such ideas are exemplary and should remain accessible in the memory of cognizable interest in the math contest. How? Well, that would be a topic for another post, but it might help to start with an inventory of key ideas for solving problems of competition - the way a chess player tirelessly studying openings, finishes, and middle game tactics. JMD in VL
wish you a happy (or not so unhappy lost) 2006
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